Energetic, Entropic, and Volumetric Effects in Nonaqueous Associated

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Energetic, Entropic, and Volumetric Effects in Nonaqueous Associated Solutions Concepcion Paz-Ramos,† Claudio A. Cerdeiri~na,*,† and Miguel Costas*,‡ † ‡

Departamento de Física Aplicada, Universidad de Vigo, Campus del Mar, As Lagoas s/n, Ourense 32004, Spain Laboratorio de Biofisicoquímica, Departamento de Fisicoquímica, Facultad de Química, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico DF 04510, Mexico

bS Supporting Information ABSTRACT: On the basis of a simple two-state association model (TSAM), a comprehensive study of thermodynamic response functions for nonaqueous associated solutions is presented. The excess isobaric heat capacities CpE(T) and excess thermal expansivities VpE (∂VE/∂T)p for a number of alcohol alkane, aminealkane, alcoholether, and alcoholalcohol mixtures have been experimentally determined at atmospheric pressure within 278.15338.15 K for CpE and 283.15333.15 K for VpE(T). A rich variety (in some cases unreported) of temperature dependences for CpE(T) but also for VpE(T) curves has been observed. This thermodynamic information has been rationalized with the TSAM, which is found to qualitatively account for all observations. Specifically, the model provides a detailed dissection of the energetic and entropic effects of association that are reflected on CpE(T), and of the volumetric effects that are echoed on VpE(T). The latter, almost unexplored to date, are found to be opposite to what is currently being conjectured for “low-temperature water”, and they stimulate further experimental studies on aqueous solutions of nonelectrolytes.

1. INTRODUCTION Statistical mechanics provides a link between thermodynamic response functions—also referred to as second-order derivatives or thermodynamic derivatives—and fluctuations. Among other relations,1 one has Cp ¼

1 ÆðδHÞ2 æ kB T 2

ð1Þ

Rp ¼

1 ÆδV δHæ ÆV ækB T 2

ð2Þ

kT ¼

1 ÆðδV Þ2 æ ÆV ækB T 2

ð3Þ

That is, the isobaric heat capacity Cp and the isothermal compressibility kT are a measure of enthalpy (Æ(δH)2æ) and volume (Æ(δV)2æ) fluctuations, respectively, while the isobaric thermal expansivity Rp quantifies the correlation between them (ÆδH δVæ). (kB is the Boltzmann constant.) Naturally, eqs 13 are consistent with the second law of thermodynamics, which implies that Cp > 0 and kT > 0 while Rp is unrestricted in sign. The local energetic, entropic, and volumetric effects associated with intermolecular interactions must manifest in the above response functions, with hydrogen bonding being an example.2,3 Indeed, the formation/disruption of hydrogen bonds in the r 2011 American Chemical Society

liquid state is accompanied by local changes in energy, entropy, and volume that contribute to δH and δV in eqs 13. Let us illustrate this for water. The enhanced “quality” of its threedimensional hydrogen-bond network as temperature is decreased—especially down to the deeply supercooled region— is accompanied by a decrease in local enthalpy, entropy, and density,4 so contributing positively to Cp and kT and negatively to Rp (Figure 1). At sufficiently higher temperatures, a fraction of hydrogen bonds are broken so that monomers, dimers, and larger multimers are present. At such temperatures the fluid resides in a low-density state, implying that formation of hydrogen bonds when O and H atoms concur in space decreases the enthalpy and entropy of the system but results in a local density increase. On the other hand, while energetic and entropic effects of H-bonding are also important in other associated liquids such as alcohols (see, e.g., refs 2, 3, 11, and 1318), experimental data give no indication of negative ÆδH δVæ (Figure 1). Since the enhancement of H-bond structure as the temperature is lowered results, as for water, in an enthalpy decrease, one may infer—from eq 2— that volumetric effects are at both high and low temperatures like those for “high-temperature” water. Clearly, that an alkyl chain (that is, a number of nonpolar chemical units) is bonded to the

Received: March 21, 2011 Published: June 27, 2011 9626

dx.doi.org/10.1021/jp205109p | J. Phys. Chem. B 2011, 115, 9626–9633

The Journal of Physical Chemistry B

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which escape from the basic nature of the model. Nevertheless, as will be shown, a qualitative analysis provides useful insights into energetic, entropic, and volumetric effects associated with H-bonding. The paper is organized as follows. In section 2 we describe the experiments. The discussion of the results and the corresponding analysis of energetic, entropic, and volumetric effects in terms of the model—which is described in the Appendix—are contained in section 3. Section 4 includes some concluding remarks.

2. EXPERIMENTAL METHODS

Figure 1. Isobaric heat capacity Cp, isobaric thermal expansivity Rp, and isothermal compressibility kT of water (4), methanol (]), and 3-ethyl3-pentanol (O) as a function of temperature T at atmospheric pressure. The left y-axes set the scales for water and methanol, and the right ones correspond to 3-ethyl-3-pentanol. Data for water were taken from refs 5 (Cp), 6 (Rp), and 7 (kT); those for methanol were taken from refs 8 (Cp), 9 (Rp), and 10 (kT); and those for 3-ethyl-3-pentanol were taken from refs 11 (Cp) and 12 (Rp and kT).

hydroxyl group of the alcohol causes this markedly different behavior. The study of response functions for liquid solutions opens a number of avenues to examine the features of association because the excess isobaric heat capacity CEp (T) is basically driven by its associational part.11,1928 This has been exploited in previous works11,25 to analyze CEp (T) for alcohol solutions, thereby enabling the identification and characterization of various interesting features of that property for inert and noninert solvents (here, noninert solvents refer to nonassociated liquids that are proton acceptors and thus capable of H-bonding with alcohols). Such a task was accomplished with the aid of a two-state association model (TSAM).11,13,25 To provide a fuller account of association effects in nonaqueous associated systems, we present here a comprehensive set of CEp (T) measurements (at atmospheric pressure) for three types of binary liquid mixtures, namely, associatedinert, associatednoninert, and associatedassociated. In addition, data for the temperature derivative of the excess volume or excess thermal expansivity VEp (∂VE/∂T)p, for which very little information and knowledge have been accumulated, are provided. All this information is discussed and qualitatively analyzed by use of the TSAM. Although the model allows obtaining information at a quantitative level when applied to the Cp of pure associated liquids,11,13 such a study does not seem pertinent for the case of binary mixtures: excess properties are small in magnitude so that their behavior cannot be quantitatively captured without a proper accounting of subtle phenomena (e.g., H-bond cooperativity),

2.1. Studied Systems. Experiments were planned using the following criteria: (i) in order to avoid the effect of orientational order between segments of linear alkyl chains of neighboring molecules,2931 liquids (self-associated and solvent) with branched alkyl chains have been selected; (ii) one alcohol and one amine were employed as self-associated liquids with a view to analyze the effect of the strength of H-bonding; and (iii) branched alkanes and mono- and polyethers were employed as inert and noninert solvents, respectively. The studied systems (with the number of measured compositions specified in parentheses) are 3M3Pbr-C8 (9), 3M3Pbr-C12 (6), 3M3Pbr-C16 (9), CxNH2br-C16 (12), 3M3Pdiglyme (3), 3M3Ptriglyme (3), 3M3Ptetraglyme (3), 3M3PDBE (3), and 3M3P1-propanol (10). The meanings of the abbreviated compound names are as follows: 3M3P 3-methyl-3-pentanol; br-C8 2,3,4-trimethylpentane; br-C12 2,2,4,6,6-pentamethylheptane; br-C16 2,2,4,4,6,8,8-heptamethylnonane; CxNH2 cyclohexylamine; DBE dibutyl ether. 2.2. Materials. The employed liquids were purchased from Fluka [n-heptane* (99.8%), toluene* (99.8%), water* (99.9%), and br-C12 (99%)] and from Aldrich [n-decane* (99%), cyclohexane* (99.9%), 1-propanol (99.4%), 1-butanol* (99.7%), 2-butanol* (99.4%), br-C8 (98%), br-C16 (98%), 3M3P (99%), CxNH2 (99%), diglyme (99%), triglyme (99%), tetraglyme (99%), and DBE (99%)]. The stated purities are expressed on a molar basis, and the liquids marked with an asterisk were used as calibrating substances. Polar liquids, namely, water, 1-butanol, 2-butanol, 3M3P, CxNH2, diglyme, triglyme, tetraglyme, and DBE, were dried using Fluka 4 nm molecular sieves. All liquids were degassed prior to preparation of mixtures, which were made by weighing using a Mettler AE-240 balance. The estimated accuracy in the mole fraction is (0.0001. 2.3. Instrumentation and Procedures. Data for the excess heat capacity CEp were obtained by combining heat capacities per unit volume CpV1 and densities F for pure liquids and mixtures (CEp Cp x1C1p x2C2p, with superscripts “1” and “2” referring to the pure liquids). Densities were measured with an AntonPaar vibrating-tube densimeter, which was calibrated by using n-heptane, cyclohexane, toluene, and Milli-Q water as density standards. Heat capacities per unit volume were determined with a Micro DSCII calorimeter from Setaram. This apparatus and the experimental technique have been described previously.32,33 n-Heptane, n-decane, 1-butanol, and 2-butanol were employed as calibrating liquids, and their Cp values were taken from the literature.8 Working in an up-scan mode at a 0.25 K 3 min1 rate, the uncertainty in CEp data was found to be (0.1 J 3 mol1 3 K1 (see ref 33 for a detailed explanation). The determination of excess thermal expansivities VEp entails, first, that of Vp (∂V/∂T)p for pure liquids and mixtures from F(T) data. Such a task was accomplished by means 9627

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Figure 3. Excess isobaric heat capacities CEp as a function of temperature T for x3M3P(1 x)br-C12. (O) x = 0.0241; (0) x = 0.4745; (4) x = 0.9487.

Figure 2. (a) Temperature dependence of the associational part of the isobaric heat capacity Cassoc,E (T) as obtained from eq A.15. The p associational excess isobaric heat capacities Cassoc,E (T) in panels bd p result from two such peaks (pure and mixture), as eq A.17 indicates. Values for ΔH ΔHM = ΔHP and ΔS are given in J 3 mol1 and J 3 mol1 3 K1, respectively.

of an incremental procedure described in detail elsewhere.34 Then, VEp Vp x1V1p x2V2p was computed; its uncertainty was (0.1 cm3 3 mol1 3 K1.34 Data were obtained with a step of 1 K within 278.15338.15 K for CEp and 283.15333.15 K for VEp . They have been deposited as Supporting Information.

3. RESULTS AND DISCUSSION 3.1. Qualitative Model Predictions. Figure 2 illustrates the model predictions—a detailed description of the model can be found in the Appendix—for the associational part of CEp (i.e., the second term in the right-hand side of eq A.18). As Figure 2a (T) results from the balance of two association indicates, Cassoc,E p (T) and Cassoc,P (T). Curves peaks corresponding to Cassoc,M p p shown in Figure 2bd correspond to particular sets of parameters, which, as we shall see, have been chosen for convenience. By comparing eqs A.18A.20, one infers an analogous qualita(T) and Vassoc,E (T). The magnitude of tive picture for Vassoc,E p T association effects is modulated by the prefactors ΔH2, ΔH ΔV, and ΔV2. As defined in the Appendix, ΔH and ΔV are the (molar) enthalpy of dissociation and (molar) volume of dissociation, respectively. Such quantities are closely related to the association energies and to the volume changes resulting from association. They are the crucial model parameters together with the dissociation entropy ΔS, obviously a measure of entropic effects involved in association processes. We will assume (see the Appendix) that ΔH, ΔS, and ΔV do not depend on temperature and pressure. Moreover, the nature of the nonassociated liquid (especially if noninert) and composition may affect the capability of a given molecule to H-bond with its neighbors as well as the hydrogen-bond energy and volume themselves, so that ΔHM(x) 6¼ ΔHP, ΔSM(x) 6¼ ΔSP, and ΔVM(x) 6¼ ΔVP. In what follows, we will show experimental data of associatedinert liquid mixtures, associatednoninert liquid mixtures,

Figure 4. Excess isobaric heat capacities CEp as a function of temperature T for xCxNH2(1 x)br-C16. (O) x = 0.2927; (0) x = 0.4960; (]) x = 0.4960. Filled squares are literature data34 for x(n-dodecane) (1x)cyclohexane with x = 0.4662. Lines are only intended to guide the eye.

and associatedassociated liquid mixtures. Such information will be compared qualitatively to that given in Figure 2. The justification and usefulness of such an approach have been provided at the end of the Introduction. 3.2. Energetic and Entropic Effects for AssociatedInert Liquid Mixtures. For associatedinert liquid mixtures one may expect that ΔHM ≈ ΔHP ΔH is a good approximation. On the other hand, since self-association in the mixture implies bringing molecules together over longer distances than for the pure associated liquid, ΔSM(x) > ΔSP (i.e., dΔSM/dx < 0). Assuming that CEp (T) is dominated by its associational contribution (which is fairly justified11), the TSAM then predicts that CEp (T) will vary with T as shown in Figure 2b,c. Concretely, for a given ΔH value, the observed CEp (T) within the fixed experimental temperature window must be seen to be displaced to the right as ΔSM(x) is increased (see Figure 2b). These predictions are corroborated by data for 3M3Palkane mixtures; the data for mixtures containing br-C12 shown in Figure 3 are a representative example of the behavior for the remaining systems. To our knowledge, experimental evidence of associatedinert liquid mixtures whose CEp (T) displays a minimum, such as those corresponding to the high-temperature region in Figure 2c, has not been previously reported. From Figure 2c, one infers that CEp (T) curves are displaced toward lower temperatures as the ΔH value is 9628



Figure 5. Excess isobaric heat capacities CEp as a function of temperature T for (a) x3M3P(1 x)diglyme [(O) x = 0.2649; (0) x = 0.5189; (4) x = 0.7519] and (b) x3M3P(1 x)DBE [(O) x = 0.2525; (0) x = 0.4747; (4) x = 0.7438].

decreased. To corroborate this TSAM prediction, it is then important to examine mixtures containing primary amines since they are characterized by a lower self-association energy (≈10 kJ 3 mol1)35 than that for alcohols (≈ 20 kJ 3 mol1).36 Figure 4 shows that CxNH2br-C16 mixtures display the predicted behavior. Beyond that, it is interesting to note that amines fall in between alcohols and long-chain normal alkanes. For the latter, it has been shown that heat capacities are sensitive to the orientation of chain segments between neighboring molecules.2931 Since orientational order embodies small energies (as compared, for instance, to self-association energy in alcohols), it seems natural that CEp (T) covers the very-high-temperature limit of the curves of Figure 2b,c. Figure 4 shows that this is indeed the case for a n-dodecanecyclohexane mixture. 3.3. Energetic and Entropic Effects for Associated Noninert Liquid Mixtures. When an alcohol is mixed with a liquid that cannot self-associate but acts as a proton acceptor, say a noninert liquid, hydrogen bonding between alcohol molecules competes with alcoholnoninert cross association. For an equimolar mixture, cross association may dominate when the fraction of association sites in the proton acceptor is sufficiently larger than that in the alcohol, with this effect being amplified as the proton acceptor mole fraction is increased. In the limiting case where the concentration of proton acceptor groups is so large that most alcohol molecules are H-bonded to one such group, ΔHM must take on the characteristic value of the cross H-bond, and since more proton acceptor sites to hydroxyl groups are available in the mixture than in the pure fluid, one may have ΔSM < ΔSP. In fact, this behavior has been found previously for alcoholester solutions.11 For the 3M3Pglyme mixtures studied in this work we may suppose37 ΔHM ≈ ΔHP while ΔSM < ΔSP, with the latter assumption being supported by the values for the fraction of proton acceptor groups in each molecule38 (namely, ϕPA = 1/3 for * = 1/7 for 3M3P). It is on this basis that qualitative glymes and ϕPA (T) are depicted in Figure 2d, predictions of the TSAM for Cassoc,E p and they are verified by the experimental CEp (T) data in Figure 5a. (We only show data for 3M3Pdiglyme mixtures; the behavior displayed by mixtures with triglyme and tetraglyme is fully analogous mainly because these three glymes are characterized by ϕPA = 1/3.) On the other hand, for the 3M3PDBE binary system the above rationale indicates that CEp (T) may approach the “alcoholinert-like” behavior since ϕPA = 1/9 for DBE. This is what is actually observed in Figure 5b.

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Figure 6. Excess isobaric heat capacities CEp as a function of (a) temperature T and (b) composition x for x3M3P(1 x)1-propanol. Compositions in (a) are (O) x = 0.2841, (0) x = 0.5944, and (4) x = 0.9510. Temperatures in (b) are (b) 278.15 K, (9) 298.15 K, (2) 318.15 K, and (1) 338.15 K. Solid lines in (b), which are intended to guide the eye, are the fitted values to the RedlichKister-type equation i CEp = x(1 x)∑ni=1 Ai(2x 1)i1/∑m i=1 Bi(2x 1) .

3.4. Energetic and Entropic Effects for Associated Associated Liquid Mixtures. Another interesting type of

mixtures would be that where the noninert or proton acceptor is also a self-associated liquid (e.g., alcoholalcohol mixtures). (T) results from three association peaks (see Since Cassoc,E p Appendix), the expected phenomenology is more diverse. Figure 6a evidences this fact for 3M3P1-propanol mixtures at three concentrations. Standing out is the unusual, previously unseen, minimum with CEp > 0 that is encountered for one of them. Furthermore, the crossing between the various CEp vs x curves of Figure 6b indicates that the type of temperature dependence changes from one composition to another, thus revealing that the balance among the various relevant contributions to CEp is a delicate one. As observed for other alcohol alcohol mixtures,24 CEp vs x curves are W-shaped. Such behavior has been frequently observed—and extensively discussed—for mixtures containing a polar (nonassociated) liquid and a nonpolar one (usually an alkane). The general consensus39,40 is that they mark the prelude of liquidliquid demixing at lower temperatures, with concentration fluctuations at intermediate compositions being responsible for the positive part of the CEp increase at intermediate compositions. This is unlikely to be a plausible picture for alcoholalcohol mixtures. Instead, the W-shape arises from the above-noted delicate balance between associational contributions to CEp . Therefore, no liquidliquid phase transition is expected at lower temperatures, all the more so because both liquids are of quite similar physicochemical nature. 3.5. Volumetric Effects. In Figure 7, which shows VEp (T) for associatedinert liquid mixtures, it is seen that trends are quite similar to those encountered for CEp (T): specifically, various portions of the curves of Figure 2a,b are covered, with basically (T) with x. That is consistent, in the the same evolution of VE,assoc p context of the TSAM, with ΔVM ≈ ΔVP ΔV > 0 (see eq A.19). Overall, for VEp (T) the temperature dependence is smoother than (T), implying that contributions arising from nonfor CE,assoc p specific interactions and packing effects (see, e.g., ref 41) play a significant role. In fact, previous estimations for alcohol solutions indicate that ΔV is a small positive parameter on the order of a few cm3 mol1.11,42 On choosing ΔV = 5 cm3 mol1, one obtains 9629



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Figure 9. Excess isobaric thermal expansivities VEp as a function of (a) temperature T and (b) composition x for x3M3P(1 x)1-propanol. Compositions in (a) are (O) x = 0.0708, (0) x = 0.5002, and (4) x = 0.9510. Temperatures in (b) are (b) 283.15, (9) 298.15, (2) 318.15, and (1) 333.15 K. Lines are only intended to guide the eye; in (b) they correspond to the fitted values to the RedlichKister-type equation i VEp = x(1 x)∑ni=1 Ai(2x 1)i1/∑m i=1 Bi(2x 1) .

Figure 7. Excess isobaric thermal expansivities VEp as a function of temperature T for (a) x3M3P(1 x)br-C8 [(O) x = 0.0506; (0) x = 0.2959; (4) x = 0.7442], (b) x3M3P(1 x)br-C12 [(O) x = 0.1074; (0) x = 0.5010; (4) x = 0.9504], (c) x3M3P(1 x)br-C16 [(O) x = 0.1495; (0) x = 0.2994; (4) x = 0.9502], and (d) xCxNH2(1 x)brC16 [(O) x = 0.0909; (0) x = 0.4960; (4) x = 0.9525]. Lines are only intended to guide the eye.

Figure 8. Excess isobaric thermal expansivities VEp as a function of temperature T for (a) x3M3P(1 x)diglyme [(O) x = 0.2649; (0) x = 0.5189; (4) x = 0.7519] and (b) x3M3P(1 x)DBE [(O) x = 0.2525; (0) x = 0.4747; (4) x = 0.7438]. Lines are only intended to guide the eye.

from eq A.19 typical values for Vassoc,E ranging from 5 to 10 cm3 p 1 1 kK mol , which are reasonably consistent with experimental observations (see Figure 7). On the other hand, the small ; i.e., by setting ΔV = ΔV value results in negligible Vassoc,E T 5 cm3 mol1 in eq A.20, one finds that the association contribution is 0.1% of VET (for which typical values range around a few cm3 mol1 GPa1 12). Data for alcoholnoninert and alcoholalcohol mixtures (Figures 8 and 9) reinforce the above conclusions; namely, they are consistent with ΔV > 0 and

small, so that the temperature dependence is smoother than that of CEp . In summary, we have found that VEp (T) is sensitive to association. Remarkably, the result ΔV > 0 indicates that the formation of H-bonds in these mixtures is accompanied by a local volume decrease. Such volumetric effects are similar to those for pure alcohols, and hence they are different from those described in the Introduction for low-temperature water.

4. CONCLUSION AND OUTLOOK The study of thermodynamic response functions for nonaqueous associated solutions of nonelectrolytes was initiated in the 1980s at McGill University under the guidance of Donald Patterson (see refs 1924). The present work culminates a series of papers on this topic, in which the “two-state” idea, originally envisioned in ref 20, has been exploited. Here, we have restricted ourselves to a qualitative treatment. A quantitative analysis would need more refined models and detailed simulations (see ref 17) that should provide a proper connection between structure and response functions. The implications of this work for aqueous solutions of nonelectrolytes might be important. Much theoretical knowledge on water and aqueous solutions has been gained during past decades.43 Particularly, progress toward elucidating intricate phenomena such as hydrophobicity and amphiphilic selfassembly44 as well as water’s anomalous thermodynamics4 has been achieved. On the other hand, existing experimental studies of response functions for aqueous solutions are rather incomplete in that they cover narrow temperature ranges (usually, data at a single temperature are reported, but see, e.g., ref 45 for early work and ref 46 for a recent account). It turns out that further experiments on the temperature and, perhaps, pressure dependence of thermodynamic response functions for aqueous solutions could enhance the actual understanding on the structure and thermodynamics of such systems. ’ APPENDIX Within the mean-field approximation, the isothermal isobaric partition function Δ of a fluid consisting of N identical 9630



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molecules is expressed as Δ¼

1 ðΔpart ÞN N!

ðA.1Þ

where Δpart stands for the partition function of a particle. We now assume that Δpart can be split as Δpart ¼ ðΔpart ÞAi þ ðΔpart ÞA

ðA.2Þ

where (Δpart)Ai and (Δpart)A represent sums over states in which an individual particle belongs to an associated species or exists as a monomer, respectively. One may think of an hypothetical fluid, A, in which associational degrees of freedom are frozen, and likewise for a hypothetical fully associated fluid, Ai, with frozen monomeric degrees of freedom. Following eq A.1, we have ΔA ¼ ΔA i

1 ðΔpart ÞNA N!

ðA.3Þ

Ai

Ai

S H þ ln N 1 ¼ exp R RT A

A

S H þ ln N 1 R RT

!

! ðA.6Þ

In eqs A.5 and A.6 HAi, HA, SAi, and SA represent the molar enthalpies and molar entropies of the hypothetical fluids. Now, substitution into eq A.2 yields ! GA ΔG ðA.7Þ Δpart ¼ exp 1 þ exp RT RT where we have used the thermodynamic identity G = H TS. The quantity ΔG GA GAi = ΔH TΔS may be seen as the Gibbs energy of dissociation for a mole of particles, with ΔH = HA HAi and ΔS = SA SAi standing for the molar enthalpy and the molar entropy of dissociation, respectively. On employing G = RT ln Δ, one readily finds

Vp ¼ VpA þ

ΔH ΔV expðΔH=RT þ ΔS=RÞ RT 2 ½expðΔH=RTÞ þ expðΔS=RÞ2

ðA.13Þ

VT ¼ VTA þ

ΔV 2 expðΔH=RT þ ΔS=RÞ RT ½expðΔH=RTÞ þ expðΔS=RÞ2

ðA.14Þ

The second term on the right-hand side of each of the eqs A.12A.14, which is the association contribution to each of those response functions, is the same for all of them except for the prefactors ΔH2, ΔH ΔV, and ΔV2. In accord with explanatory comments in the Introduction, eqs A.12A.14 explicitly include parameters characterizing energetic, entropic, and volumetric effects of H-bonding (i.e., ΔH, ΔS, and ΔV), and consistently reflect the content of eqs 13. By adopting—as is frequently done—the ideal gas as a res reference state, one has Cp = Cigp + Cres p , where Cp (the residual heat capacity) is that part of Cp arising from intermolecular interactions. Beyond that, the Cres p of an associated liquid may be split into an associational part and a “nonspecific” one arising from assoc = Cns . Since dispersive interactions etc., i.e., Cres p p + Cp A ig ns Cp = Cp + Cp , eq A.12 turns into ns Cres p ¼ Cp þ R

GðT, pÞ ¼ GA ðT, pÞ ΔHðT, pÞ ΔSðT, pÞ RT ln 1 þ exp RT R

ðA.8Þ

where the explicit dependence of the macroscopic quantities ΔH and ΔS on T and p has been noted. To leading order, ΔH and ΔS are given by ! ∂ΔH ΔHðT, pÞ ¼ ΔH0 þ ΔCp ðT T0 Þ þ ðp p0 Þ ∂p T

ðA.9Þ T ΔSðT, pÞ ¼ ΔS0 þ ΔCp ln þ T0

ðA.12Þ

ðA.5Þ

! ∂ΔS ðp p0 Þ ∂p T

ðA.10Þ

ΔH RT

2

expðΔH=RT þ ΔS=RÞ ½expðΔH=RTÞ þ expðΔS=RÞ2

Cp ¼ CAp þ R

ðA.4Þ

where

ðΔpart ÞA ¼ exp

T

Given these rather general expressions, we now anticipate our assumption that ΔH, ΔS, and ΔV are constant. As shown previously,11,13 this assumption is a reasonable approximation within moderately small temperature and pressure ranges. Thus, the response functions are derived from eq A.8 via standard formulas, namely, Cp T(∂2G/∂T2)p, Vp (∂V/∂T)p = ∂2G/∂p∂T, and VT (∂V/∂p)T = (∂2G/∂p2)T to get

1 ¼ ðΔpart ÞNA i N!

ðΔpart ÞA i

where a reference state, denoted by subscript “0”, has been introduced, while ΔCp T(∂ΔS/∂T)p = (∂ΔH/∂T)p. Furthermore, one can define the volume of dissociation as ΔV (∂ΔG/∂p)T, and express it as ∂ΔV ΔV ðT, pÞ ¼ ΔV0 þ ðT T0 Þ ∂T T ! ∂ΔV ðp p0 Þ ðA.11Þ þ ∂p

2 ΔH expðΔH=RT þ ΔS=RÞ RT ½expðΔH=RTÞ þ expðΔS=RÞ2

ðA.15Þ

The extension of the model to a binary mixture composed by an associated liquid and one whose molecules are unable to self-associate, inert, or noninert (see Introduction) leads to the following expression for excess magnitudes: Y E ¼ Y ns, E þ Y assoc, E ¼ Y ns, E þ Y assoc, M xY assoc, P ðA.16Þ where Y can be Cp, Vp, or VT and x is the mole fraction of selfassociated liquid, whereas Yassoc,M and Yassoc,P refer to the associational contribution for 1 mol of mixture and 1 mol of pure fluid, respectively, and Yns,E = YA,E. In a mixture, only a fraction of the total number of molecules (the mole fraction x) is responsible for association effects, so Yassoc,M = xYassoc,M*, where Yassoc,M* denotes the associational contribution for a mixture (with mole 9631



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fraction x) containing 1 mol of molecules capable of self-association. Therefore Y E ¼ Y ns, E þ Y assoc, E ¼ Y ns, E þ x½Y

assoc, M

Y assoc, P

ðA.17Þ and E CEp ¼ Cns, p

8 ! < ΔH M 2 expðΔH M =RT þ ΔSM =RÞ þ xR : RT ½expðΔH M =RTÞ þ expðΔSM =RTÞ2 ) !2 ΔH P expðΔH P =RT þ ΔSP =RÞ RT ½expðΔH P =RTÞ þ expðΔSP =RTÞ2

ðA.18Þ VpE ¼ Vpns, E þ xR

(

ΔH M ΔV M expðΔH M =RT þ ΔSM =RÞ 2 ½expðΔH M =RTÞ þ expðΔSM =RTÞ2 ðRTÞ

ΔH P ΔV P expðΔH P =RT þ ΔSP =RÞ 2 ðRTÞ ½expðΔH P =RTÞ þ expðΔSP =RTÞ2

ðA.19Þ VTE ¼ VTns, E

8 ! < ΔV M 2 expðΔH M =RT þ ΔSM =RÞ þ xRT : RT ½expðΔH M =RTÞ þ expðΔSM =RTÞ2 ) !2 ΔV P expðΔH P =RT þ ΔSP =RÞ RT ½expðΔH P =RTÞ þ expðΔSP =RTÞ2

ðA.20Þ Finally, for the case of a mixture of two associated liquids, the associational part in eq A.16 must be replaced by Y assoc, E ¼ Y assoc, M xY assoc, P1 xY assoc, P2

ðA.21Þ

where subscripts “1” and “2” refer to the two pure associated liquids. Hence, Yassoc,E results from three association peaks.

’ ASSOCIATED CONTENT

bS

Supporting Information. Microsoft Excel files contain the experimental data for CEp (T) and VEp (T) for each system. Although not discussed in this work, data for the excess volume VE(T) are also deposited for completeness. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (C.A.C.); [email protected]. mx (M.C.).

’ ACKNOWLEDGMENT The research of C.P.-R. was supported by the Spanish Ministry of Education (Grant AP-2002-2764).

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ARTICLE

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Energetic, Entropic, and Volumetric Effects in Nonaqueous Associated

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